Optimal. Leaf size=267 \[ \frac{5 a^4 x^5 (587-109 a x) \left (c-\frac{c}{a x}\right )^{9/2}}{7 (1-a x)^{9/2} \sqrt{a x+1}}+\frac{70 a^3 x^4 \left (c-\frac{c}{a x}\right )^{9/2}}{(1-a x)^{5/2} \sqrt{a x+1}}-\frac{50 a^2 x^3 \left (c-\frac{c}{a x}\right )^{9/2}}{7 (1-a x)^{3/2} \sqrt{a x+1}}-\frac{15 a^{7/2} x^{9/2} \left (c-\frac{c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}+\frac{10 a x^2 \left (c-\frac{c}{a x}\right )^{9/2}}{7 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 x \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{9/2}}{7 \sqrt{a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.242904, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6134, 6129, 98, 150, 143, 54, 215} \[ \frac{5 a^4 x^5 (587-109 a x) \left (c-\frac{c}{a x}\right )^{9/2}}{7 (1-a x)^{9/2} \sqrt{a x+1}}+\frac{70 a^3 x^4 \left (c-\frac{c}{a x}\right )^{9/2}}{(1-a x)^{5/2} \sqrt{a x+1}}-\frac{50 a^2 x^3 \left (c-\frac{c}{a x}\right )^{9/2}}{7 (1-a x)^{3/2} \sqrt{a x+1}}-\frac{15 a^{7/2} x^{9/2} \left (c-\frac{c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}+\frac{10 a x^2 \left (c-\frac{c}{a x}\right )^{9/2}}{7 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 x \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{9/2}}{7 \sqrt{a x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6134
Rule 6129
Rule 98
Rule 150
Rule 143
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{9/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{e^{-3 \tanh ^{-1}(a x)} (1-a x)^{9/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^6}{x^{9/2} (1+a x)^{3/2}} \, dx}{(1-a x)^{9/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1-a x}}{7 \sqrt{1+a x}}-\frac{\left (2 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^4 \left (\frac{25 a}{2}-\frac{5 a^2 x}{2}\right )}{x^{7/2} (1+a x)^{3/2}} \, dx}{7 (1-a x)^{9/2}}\\ &=\frac{10 a \left (c-\frac{c}{a x}\right )^{9/2} x^2}{7 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1-a x}}{7 \sqrt{1+a x}}-\frac{\left (4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^3 \left (-\frac{375 a^2}{4}-\frac{25 a^3 x}{4}\right )}{x^{5/2} (1+a x)^{3/2}} \, dx}{35 (1-a x)^{9/2}}\\ &=-\frac{50 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{10 a \left (c-\frac{c}{a x}\right )^{9/2} x^2}{7 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1-a x}}{7 \sqrt{1+a x}}-\frac{\left (8 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^2 \left (\frac{3675 a^3}{8}+\frac{825 a^4 x}{8}\right )}{x^{3/2} (1+a x)^{3/2}} \, dx}{105 (1-a x)^{9/2}}\\ &=\frac{70 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt{1+a x}}-\frac{50 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{10 a \left (c-\frac{c}{a x}\right )^{9/2} x^2}{7 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1-a x}}{7 \sqrt{1+a x}}-\frac{\left (16 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x) \left (-\frac{21225 a^4}{16}-\frac{8175 a^5 x}{16}\right )}{\sqrt{x} (1+a x)^{3/2}} \, dx}{105 (1-a x)^{9/2}}\\ &=\frac{5 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^5 (587-109 a x)}{7 (1-a x)^{9/2} \sqrt{1+a x}}+\frac{70 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt{1+a x}}-\frac{50 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{10 a \left (c-\frac{c}{a x}\right )^{9/2} x^2}{7 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1-a x}}{7 \sqrt{1+a x}}-\frac{\left (15 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{9/2}}\\ &=\frac{5 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^5 (587-109 a x)}{7 (1-a x)^{9/2} \sqrt{1+a x}}+\frac{70 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt{1+a x}}-\frac{50 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{10 a \left (c-\frac{c}{a x}\right )^{9/2} x^2}{7 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1-a x}}{7 \sqrt{1+a x}}-\frac{\left (15 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{9/2}}\\ &=\frac{5 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^5 (587-109 a x)}{7 (1-a x)^{9/2} \sqrt{1+a x}}+\frac{70 a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt{1+a x}}-\frac{50 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt{1+a x}}+\frac{10 a \left (c-\frac{c}{a x}\right )^{9/2} x^2}{7 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1-a x}}{7 \sqrt{1+a x}}-\frac{15 a^{7/2} \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}\\ \end{align*}
Mathematica [C] time = 10.0585, size = 234, normalized size = 0.88 \[ -\frac{c^4 \sqrt{c-\frac{c}{a x}} \left (-\frac{7168 (-a x (a x+1))^{5/2} (a x-1)^4 \text{HypergeometricPFQ}\left (\left \{-\frac{3}{2},2,2,2,\frac{5}{2}\right \},\left \{1,1,1,\frac{7}{2}\right \},-a x\right )}{(a x+1)^{3/2}}+\sqrt{-a x (a x+1)} \left (70000 a^8 x^8-214760 a^7 x^7+165830 a^6 x^6+84329 a^5 x^5-375805 a^4 x^4-59750 a^3 x^3+34100 a^2 x^2-12955 a x+3091\right )+105 \left (101 a^5 x^5+209 a^4 x^4-54 a^3 x^3-54 a^2 x^2+81 a x-27\right ) \sin ^{-1}\left (\sqrt{-a x}\right )\right )}{896 a^4 x^3 \sqrt{-a x (a x+1)} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.158, size = 227, normalized size = 0.9 \begin{align*} -{\frac{{c}^{4}}{14\,{x}^{3} \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 14\,\sqrt{- \left ( ax+1 \right ) x}{a}^{11/2}{x}^{5}+105\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{5}{a}^{5}+3510\,{a}^{9/2}\sqrt{- \left ( ax+1 \right ) x}{x}^{4}+105\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{4}{a}^{4}+1440\,{a}^{7/2}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}-220\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+40\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}-4\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ) \sqrt{-{a}^{2}{x}^{2}+1}{a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.19902, size = 848, normalized size = 3.18 \begin{align*} \left [\frac{105 \,{\left (a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \,{\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{28 \,{\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, \frac{105 \,{\left (a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{14 \,{\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]